1. Holographic Magnetism from General Relativity
Kobe University，
Holographic Magnetism from General Relativity
Rong-Gen Cai (蔡荣根）
Institute of Theoretical Physics
Chinese Academy of Sciences
Yukawa Institute for Theoretical Physics
Kyoto University
Refs: arXiv: , , , ,
, , , ,

2. 2015: GR100
GR is a theory about
space and time!
引力（时空）的本质？
1500年前，奥古斯丁
的名言：时间是什么？

3. It is more than that!
It can also tell us something which is not relevant to gravity!
This is the goal of my talk:
Magnetism from GR!
The idea is that the gravity in AdS bulk is equivalent to a
CFT at the AdS boundary, AdS/CFT correspondence,
a realization of the holographic principle.

4. Black hole is a window to quantum gravity
1、Introduction: holographic principle
Black hole is a window to quantum gravity
Thermodynamics of black hole
S.Hawking, 1974, J. Bekenstein, 1973

5. Entropy in a system with surface area A :S<A/4G
Holography of Gravity
Entropy in a system with surface area A :S<A/4G
A system with gravity in d+1 dimensions can be described by a theory without gravity in d dimension!
(G. t’ Hooft)
(L. Susskind)
The world is a hologram？

6. Why GR?
The planar black hole with AdS radius L=1:
where:
Temperature of the black hole:
Energy of the black hole:
Entropy of the black hole:
The black hole behaves like a thermal gas
in 2+1 dimensions in thermodynamics!

7. Topology theorem of black hole horizon:

8. AdS/CFT correspondence（1997, J. Maldacena）:
“Real conceptual change
in our thinking about Gravity.”
(E. Witten, Science 285 (1999) 512

9. AdS/CFT dictionary : Here in the bulk:
the boundary value of the field propagating in the bulk
in the boundary theory:
the source of the operator dual to the bulk field

10. Quantum field theory
in d-dimensions
operator Ο
boundary
quantum gravitational theory
in (d+1)-dimensions
dynamical field φ
bulk
( , S. Hartnoll)

11. AdS/CFT correspondence： gravity/gauge field
2) different spacetime dimension
3) weak/strong duality
4) classical/quantum
Applications in various fields:
low energy QCD (AdS/QCD),
condensed matter theory (AdS/CMT)
e.g., holographic superconductivity
(non-) Fermion fluid

12. Holographic magnetism:
Paramagnetism-Ferromagnetism Phase Transition in a Dyonic Black Hole
Phys. Rev. D 90, (2014) (Rapid Communication)
2) Model for Paramagnetism/antiferromagnetism Phase Transition
Phys. Rev. D 91, (2015)
3) Coexistence and competition of ferromagnetism and p-wave superconductivity in
holographic model
Phys. Rev. D 91, (2015)
4) Holographic model for antiferromagnetic quantum phase transition induced by
magnetic field
Phys. Rev. D 92, (2015)
5) Antisymmetric tensor field and spontaneous magnetization in holographic duality
Phys. Rev. D 92, (2015)
6) Holographic antiferromganetic quantum criticality and AdS_2 scaling limit
Phys. Rev. D 92, (2015)
7) Massive 2-form field and holographic ferromagnetic phase transition
JHEP 1511 (2015) 021
8) Insulator/metal phase transition and colossal magnetoresistance in holographic model
Phys.Rev.D92 (2015)106002
9) Introduction to Holographic Superconductor Models
Sci. China Phys. Mech. Astron. 58 (2015)

13. Outline: 1 Introduction
2 Ferromagnetism/paramagnetism phase transition
3 Antiferromagnetism/paramagnetism phase transition
4 Antiferromagnetic quantum phase transition
5 Insulator/metal phase transition and colossal
magnetoresistance effect
6 Coexistence and competition between ferromagnetism
and superconductivity
7 Summary

14. how to build a holographic model of superconductors
CFT CFT/AdS Gravity
Global symmetry Abelian gauge field
Scalar operator Scalar field
Temperature Black hole
Phase transition High T/no hair
Low T/ hairy BH
G.T. Horowitz,

15. Holographic superconductors
Building a holographic superconductor
S. Hartnoll, C.P. Herzog and G. Horowitz, arXiv:
PRL 101, (2008)
High Temperature（black hole without hair):

16. Consider the case of m^2L^2=-2，like a conformal scalar field.
In the probe limit and A_t= Phi
At the large r boundary:
Scalar operator condensate
O_i:

18. Maxwell equation with zero momentum :
Conductivity
Maxwell equation with zero momentum :
Boundary conduction:
at the horizon: ingoing mode
at the infinity:
AdS/CFT
current
source:
Conductivity:

19. A universal energy gap: ~ 10%
BCS theory: 3.5
K. Gomes et al, Nature 447, 569 (2007)

20. Summary: The CFT has a global abelian symmetry corresponding a
massless gauge field propagating in the bulk AdS space.
Also require an operator in the CFT that corresponds to a scalar
field that is charged with respect to this gauge field..
3. Adding a black hole to the AdS describes the CFT at finite
temperature.
Looks for cases where there are high temperature black hole
solutions with no charged scalar hair, but below some critical
temperature black hole solutions with charged scalar hair and
dominates the free energy.

21. arXiv: , PRD82 (2010)
Breaking a global SU(2) symmetry representing spin into a U(1) subgroup.
The symmetry breaking is triggered by
condensation of a triplet scalar field .
This model leads to the spatial rotational
symmetry breaking spontaneously, the time reversal symmetry is not broken spontaneously in the magnetic ordered phase.

22. 2、A model for ferromagnetism/paramagnetism transition
arXiv: , PRD 90 (2014) , Rapid Comm.
The model:
Motivation is as follows: 1) time reversal symmetry and rotating symmetry are broken
2) In MFT, the order parameter is a pseudovector and is coupled to external magnetic field;
3) In weak external magnetic field, it is proportional to external magnetic field.
The reasons:
The ferromagnetic transition breaks the time reversal symmetry, spatial rotating
symmetry, but is not associated with any symmetry such as U(1), SU(2).
The magnetic moment is a spatial component of a tensor,
3)In weak external magnetic field, it is proportional to external magnetic field.

23. We are considering the probe limit, the background is
Temperature:
The ansatz:

24. The boundary condition:

25. The off-shell free energy:
Ising-like model:
arXiv:
on shell:

26. Spontaneous magnetization: B=0

27. The response to external magnetic field:
magnetic susceptibility:
Obey the Curie-Weiss Law

28. The hysteresis loop in a single magnetic domain:
When T < Tc, the magnetic moment is not single valued. The parts DE and BA are stable, which can be realized in the external field. The part CF is unstable which cannot exist in the realistic system. The parts EF and CB are metastable states, which may exist in some intermediate processes and can be observed in experiment. When the external field continuously changes, the metastable states of magnetic moment can appear.

29. 3、Faramagnetism/antiferromagnetism phase transition
arXiv:
Antiferromagnetic material does not show any macroscopic magnetic moment when external magnetic field is absent, it is still a kind of magnetic ordered material when temperature is below the Neel temperature T_N. The conventional picture, due to L.
Neel, represents a macroscopic antiferromagnetism as consisting of two sublattices, such that spins on one sublattice point opposite to that of the other sublattice. The order parameter is the staggered magnetization, as the diference between the two magnetic moments associated with the two sublattices:

30. Magnetic susceptibility:

31. Three minimal requirements to realize the holographic model
for the phase transition of paramagnetism/antiferromagnetism.
The antiparallel magnetic structure as T<T_N
The susceptibility behavior
Breaking the time reversal symm & spatial rotating symm
Our model:

32. The probe limit
The ansatz:
Define:

33. The equations of motion:
The boundary conditions:

34. The parameter constraint:
The on-shell free energy:

35. alpha_0 and beta_0 are initial values at the horizon!

37. The influence on strong external magnetic field

38. arXiv:
New model:

41. 4、Antiferromagnetic quantum phase transition

42. critical magnetic field
Er2-2xY2xTi2O7
critical magnetic field
Ex:4.2, Th:5.0
Dynamical exponent: 2

43. 5、Insulator/metal phase transition and colossal
magnetoresistance in holographic massive gravity
Some magnetic materials such as manganites
exhibit the colossal magnetoresistance effect.
Our model:
Blake, Tong and Vegh, arXiv:
Blake and Tong, arXiv:
Mefford and Horowitz, arXiv:
There is a position dependent mass
This measures the strength of inhomogeneity

44. The black brane solution:
The ansatz:
The asymptotic solution at the boundary:

45. DC conductivity:
The perturbation:
The AdS boundary:
DC resistivity:

46. By the membrane paradigm:
Iqbal and Liu, arXiv:
The DC resistivity in the strong inhomogeneity limit:

47. Numerical results:
A. Urushibara et al, PRB51 (1995)

48. 6、Coexistence between ferromagnetism and p-wave order
Coexistence between superconductivity and Ferromagnetism!

49. 6、Coexistence between ferromagnetism and p-wave order
(arXiv: )
P-wave: Einstein-Maxwell-Complex vector model:
arXiv: , JHEP 1401 (2014) 032
“A holographic p-wave superconductor model”

50. Our model:
plays a crucial role

51. The probe limit:
The ansatz:
It is found that only the following case is consistent

52. Define:

53. 1) Superconducting ferromagnet

57. 2) Ferromagnetic superconductor

60. Conclusions:
1) In the case that the ferromagnetic phase appears first,
if the interaction is attractive, the system shows the ferromagnetism and superconductivity can coexist in low temperatures. If the interaction is repulsive, the system will only be in a pure ferromagnetic state.
2) In the case that the superconducting phase appears first, the attractive interaction will lead to a magnetic p-wave superconducting phase in low temperatures. If the interaction is repulsive, the system will be in a pure p-wave superconducting phase or ferromagnetic phase when the temperature is lowered.

61. 7、Summary
1) present a holographic model for the paramagnetism
-ferromagnetism phase transition
2) realize the paramagnetism-antiferromagnetism transition
antiferromagnetic quantum phase transition
4）insulator/metal phase transition and
colossal magnetoresistance effect
5) coexistence and competition between ferromagnetism
and p-wave superconductivity

62. Thanks !